Suppose we have two matrix valued functions A, B where:
$A:\mathbb{R}^{2}\rightarrow \mathbb{R}^{nxn}$
$B:\mathbb{R}\rightarrow \mathbb{R}^{nxn}$
with $\forall \; x,y \in \mathbb{R}$:
$\frac{\delta A}{\delta x}=B(x)A(x,y)$
$\frac{\delta A}{\delta y}=-A(x,y)B(y)$
$A(x,x)=I$
Show that:
$A(x,y)A(y,z)=A(x,z)$
I'm really at a total loss here, the result seems intuitively wrong.
Derivate $A(x,y)A(y,z)$ in $y$ direction. This gives zero, using the product rule and the equations you supplied for partial derivations of $A$.
$\frac{∂}{∂y}\left(A(x, y)A(y,z)\right) = \left(\frac{∂}{∂y}A(x, y)\right)A(y,z) + A(x, y)\left(\frac{∂}{∂y}A(y,z)\right) = -A(x,y)B(y)A(y,z) + A(x, y)B(y)A(y,z) = 0$
So you can always replace $A(x,y)A(y,z) = A(x, x)A(x,z) = A(x, z)$ using $A(x, x) = I$.